1 Analysis Cohort

In this analysis, there are 150 subjects and 1205 observations. Mean (range) of follow-up time per patient was 1 (0, 2) years. Mean (range) number of observations was 8 (1, 29). The same number of patients and observations were utilized in each model below. We assume for each model that the data are missing at random, except for the joint longitudinal-survival model. The dependent variable is GLI FEV1 (% predicted) named “y” in the output below. The time scale is patients’ age at visit. Age is expressed in years. Age at CF diagnosis (in years) was included as a covariate in each model. Below are summary statistics for each variable used in the model. Once a patient was recorded as having a lung transplant, his or her data were censored from the model.

##        y               Age        Age_Diag_per_patient
##  Min.   : 10.60   Min.   : 6.01   Min.   : 0.000      
##  1st Qu.: 43.57   1st Qu.:10.94   1st Qu.: 0.100      
##  Median : 73.99   Median :18.67   Median : 0.300      
##  Mean   : 70.57   Mean   :19.47   Mean   : 2.004      
##  3rd Qu.: 94.66   3rd Qu.:24.54   3rd Qu.: 1.700      
##  Max.   :131.21   Max.   :52.12   Max.   :17.500

The Kaplan-Meier (KM) survival curve for those who died or received a lung transplant is plotted below showing the survival probability over age at visit (top graphic) and the number still alive/without lung transplant (bottom graphic).

For this scenario, we have 18 subjects (out of total 150) who subjects who died or received lung transplant over ages. The event rate is around 0.12 or 12%. The minimum time is 12.5 years old and maximum is 51 years old. All 18 subjects have different event time.

In the following report, we investigate three commonly used structures in CF FEV1 analysis. These include the linear mixed-effects model (LME), Generalized Estimating Equations (GEE) and joint modelling (JM). For the LME models, we include different combinations of random effects, splines and correlation structures as used in published CF studies (Section 2). We examine published GEE models combined with additional features for linearity/non-linearity (Section 3). We fit a joint model with the linear/non-linear structure from the LME models in Section 2.

2 Linear Mixed-Effects Model (LME)

2.1 LME Model with Random Intercept

2.1.1 Assumptions

Average evolution: A linear evolution is assumed over time.

Patient-specific evolution: A linear evolution is assumed over time. We assume that each patient may start from a different time point but all patients have the same evolution over time.

2.1.2 Model Estimation

## [1] "AIC =  9292"

## [1] "BIC =  9318"

Variance Components
	StdDev
(Intercept)	22.589837
Residual	9.014941

	Value	Std.Error	DF	t-value	p-value
(Intercept)	104.044	3.939	1054	26.410	0.000
Age	-1.589	0.184	1054	-8.620	0.000
Age_Diag	0.789	0.525	148	1.504	0.135

As the coefficient estimate for ‘Age’ is around -1.589 % predicted/year, this means for one year increase in age, FEV1 is decreasing by 1.589 (unit % predicted).

2.1.3 Model Inference

We can calculate the population-level evolution (left) and rate of change (right) over time. As we have assumed a linear evolution, the left figure shows a straight line, corresponding to a linear estimated fit. The rate of change is calculated by taking first derivative of the model equation over time. This model estimates the rate of change in FEV1 as constant over different ages as -1.589.

2.1.4 Model Diagnostics

We examined standardized residuals in terms of randomness, spread and nature of distribution.

The residuals vs fitted plot (upper left): residuals are randomly distributed over the fitted values and are clustered around 0. But we notice there are several points are apart from 0, (i.e. one point at residual round 10 and fitted value around 100). This could be due to the unbalance distribution of FEV1 or the extreme value in data.
The residuals vs age (upper right): These are randomly scattered over time (age) and the spread of the points is slightly decreasing over time. The assumption of homogeneity for the residuals of variance is roughly hold. However, we also notice few points are apart from the 0 horizontal line.
The quantile-quantile (QQ) plot (lower left): We used the QQ plot to check the normality assumption, which compares quantiles of the standardized residuals to those from a standard normal distribution. Having all points along the diagonal line (red line in plot) would suggest that the residuals are approximately normally distributed. However, in this QQ plot, there is evidence that the distribution is heavier tailed where this feature is sometimes evident when modeling longitudinal data.
The density of residuals (lower right): indicates the distribution of the residuals is slightly left skewed. These diagnostics show the model is mainly valid, but further investigation may be needed to account for the fact that the variance decreases over time and to accommodate the heavier tails.

2.2 LME Model with Random Intercept and Slope

2.2.1 Assumptions

Average evolution: A linear evolution is assumed over time.

Patient-specific evolution: A linear evolution is assumed over time. We assume that each patient may start from a different time point but all patients have different evolution over time.

2.2.2 Model Estimation

## [1] "AIC =  9287"

## [1] "BIC =  9323"

Variance Components
	StdDev
(Intercept)	14.199824
Age	0.589098
Residual	8.990464

	Value	Std.Error	DF	t-value	p-value
(Intercept)	105.611	3.578	1054	29.516	0.000
Age	-1.677	0.200	1054	-8.386	0.000
Age_Diag	0.770	0.570	148	1.351	0.179

As the estimation of coefficient for ‘Age’ is around -1.677 % predicted/year, this means for one year increase in age, FEV1 is decreasing by 1.677 (unit % predicted).

2.2.3 Model Inference

2.2.4 Model Diagnostics

The model diagnostic plots are below and they are similar to the previous pattern.

2.3 LME Model with 5-knot Natural Spline

2.3.1 Assumptions

Average evolution: A non-linear evolution (assuming cubic natural splines) is assumed over time.

Patient-specific evolution: A non-linear evolution (assuming cubic natural splines) is assumed over time. A diagonal variance-covariance matrix for the random effects are assumed. This means that the random effects are not correlated.

The knots, listed below, were chosen according to a recursive algorithm (reference: Spline Regression with Automatic Knot Selection, Vivien Goepp (2018)). There are 6 knots in total selected for the entire dataset (age: 6 years and older).

## [1] "Knots location (at age): 9.8, 13.5, 17.3, 21.4, 31.3"

2.3.2 Model Estimation

## [1] "AIC =  9237"

## [1] "BIC =  9319"

Variance Components
	StdDev
(Intercept)	16.218884
ns1	20.607789
ns2	17.685630
ns3	29.868936
ns4	31.471322
ns5	13.378872
ns6	66.527389
Residual	8.436234

	Value	Std.Error	DF	t-value	p-value
(Intercept)	99.142	3.815	1049	25.985	0.000
ns1	-13.274	6.205	1049	-2.139	0.033
ns2	-19.625	6.127	1049	-3.203	0.001
ns3	-41.742	6.972	1049	-5.987	0.000
ns4	-27.559	18.199	1049	-1.514	0.130
ns5	-127.423	30.227	1049	-4.215	0.000
ns6	-171.810	63.189	1049	-2.719	0.007
Age_Diag	0.885	0.571	148	1.551	0.123

2.3.3 Model Inference

As non-linear evolution over time is assumed, the population-level evolution (left) and rate of change (right) are varying over time. The population-level trajectory is slightly curved during the age range 6 to 30 years old and is linearly decreasing afterward. The rate of change plot shows a fast change at earlier age (around 30 years old) with a lowest decline occurred around age 20 years old.

2.3.4 Model Diagnostics

2.4 Linear Random Intercepts with Exponential Correlation

2.4.1 Assumptions

Average evolution: A linear mean evolution is assumed over time.

Patient-specific evolution: A linear evolution is assumed over time. We assume that each patient may start from a different time point but all patients have the same evolution over time. Correlation structure: exponential correlation function. This means that, for each subject, the correlation of FEV1 exponentially decays to zero with the distance between times (ages).

2.4.2 Model Estimation

## [1] "AIC =  9216"

## [1] "BIC =  9252"

Variance Components
	StdDev
(Intercept)	0.04146124
Residual	24.35877545

## Correlation Structure: Exponential spatial correlation
##  Formula: ~Age | eDWID 
##  Parameter estimate(s):
##      range     nugget 
## 19.6114439  0.1043369

	Value	Std.Error	DF	t-value	p-value
(Intercept)	103.294	4.047	1054	25.525	0.000
Age	-1.539	0.191	1054	-8.041	0.000
Age_Diag	0.736	0.528	148	1.395	0.165

As the estimation of the coefficient for ‘Age’ is around -1.539 % predicted/year, this means for one year increase in age, FEV1 is decreasing by 1.539 (unit % predicted).

2.4.3 Model Inference

2.4.4 Model Diagnostics

The residuals vs fitted plot (upper left): there is a decreasing trend in variability of residuals over the fitted value which might due to the unbalance of the response variable FEV1.
The residuals vs age (upper right): The spread of the scatter is decreasing over time and all points are roughly clustered around 0 and roughly symmetric.
The quantile-quantile (QQ) plot (lower left): is slightly heavier tailed, but is better than previous models.
The density of residuals (lower right): has a heavier right tail and not symmetric. Further investigation may be needed.

2.5 Spline Random Intercept with Exponential Correlation

2.5.1 Assumptions

Average evolution: A non-linear evolution (assuming cubic natural splines) is assumed over time.

Patient-specific evolution: A linear evolution is assumed over time and each patient may start at different time point

Correlation structure: exponential correlation function. This means that, for each subject, the correlation of FEV1 exponentially decays to zero with the distance between times (age).

2.5.2 Model Estimation

## [1] "Knots location (at age): 9.8, 13.5, 17.3, 21.4, 31.3"

## [1] "AIC =  9219"

## [1] "BIC =  9280"

Variance Components
	StdDev
(Intercept)	0.05198012
Residual	24.10691172

## Correlation Structure: Exponential spatial correlation
##  Formula: ~Age | eDWID 
##  Parameter estimate(s):
##      range     nugget 
## 19.8375331  0.1067286

	Value	Std.Error	DF	t-value	p-value
(Intercept)	99.410	4.917	1049	20.217	0.000
ns1	-12.826	7.291	1049	-1.759	0.079
ns2	-18.860	7.274	1049	-2.593	0.010
ns3	-42.782	7.441	1049	-5.749	0.000
ns4	-26.507	18.945	1049	-1.399	0.162
ns5	-119.501	30.896	1049	-3.868	0.000
ns6	-157.231	64.642	1049	-2.432	0.015
Age_Diag	0.672	0.531	148	1.267	0.207

2.5.3 Model Inference

2.5.4 Model Diagnostics

3 Linear Generalized Estimating Equations Model (GEE)

Compared with LME models, GEE models allow for weaker distributional assumptions and are more robust to covariance misspecification. In this analysis, the average evolution (fixed-effects) structure in the GEE model was selected based on the performance of all LME models. This GEE model is fitted with a linear structure to model the mean evolution and accounts for longitudinal correlation using the sandwich estimator.

3.1 Assumptions

Average evolution: A linear evolution (assuming natural cubic splines) is used over time.

Patient-specific evolution: A linear evolution is assumed over time and each patient may start at different time point.

Correlation structure: independence (working) correlation function.

3.2 Model Estimation

Coefficients:
	Estimate	Std.err	Wald	Pr(>\|W\|)
(Intercept)	100.250	4.208	567.622	0.000
Age	-1.596	0.176	82.669	0.000
Age_Diag	0.810	0.599	1.830	0.176

Estimated Scale Parameters:
	Estimate	Std.err
(Intercept)	599.319	55.692

3.3 Model Inference

3.4 Model Diagnostics

4 Non-linear Generalized Estimating Equations Model (GEE)

In this analysis, the GEE mode is fitted with a spline structure to model the mean evolution and includes an independence correlation.

4.1 Assumptions

Average evolution: A non-linear evolution (assuming cubic natural splines) is used over time.

Patient-specific evolution: A linear evolution is assumed over time and each patient may start at different time point.

Correlation structure: independence correlation function (sandwich estimator used to account for longitudinal correlation).

4.2 Model Estimation

Coefficients:
	Estimate	Std.err	Wald	Pr(>\|W\|)
(Intercept)	91.492	8.997	103.401	0.000
ns1	-3.853	13.751	0.078	0.779
ns2	-15.036	13.746	1.196	0.274
ns3	-45.308	11.940	14.399	0.000
ns4	-17.317	23.482	0.544	0.461
ns5	-108.001	33.368	10.476	0.001
ns6	-168.365	64.493	6.815	0.009
Age_Diag	0.660	0.530	1.549	0.213

Estimated Scale Parameters:
	Estimate	Std.err
(Intercept)	569.361	56.561

4.3 Model Inference

4.4 Model Diagnostics

The quantile-quantile (QQ) plot (lower left): The Q-Q plot shows that residuals approximate the unit normal distribution, which suggests that the relaxed assumptions in GEE model alleviate issues with heavy tails. The density of residuals (lower right): has a bit heavier left tail, but roughly symmetric.

5 Linear Joint Model of Longitudinal FEV1 (% predicted) and Survival (JM)

Missing data due to death or lung transplant is accounted for by jointly modeling the FEV1 process with the survival process using a Weibull model for survival. We use a Weibull baseline risk function to fit the baseline hazard in the survival sub-model. This joint model was implemented using the shared parameter framework. The shared parameters here corresponded to connecting the survival outcome with the underlying value of FEV1 and rate of change in FEV1 over time.

5.1 Assumptions

We assume that FEV1 and survival are associated over time (age). To fit the longitudinal sub-model for FEV1, we apply the LME model with linear assumption; for survival, we use the Weibull survival model.

Average evolution of FEV1: A linear evolution is assumed over time.

Patient-specific evolution of FEV1: A linear evolution is assumed over time and each patient may start at different time point (and have the same evolution).

Survival Baseline hazard: A Weibull baseline risk hazard is assumed over time at baseline.

Association: We assume the underlying value of FEV1 and rate of change over time to be associated with survival.

5.2 Model Estimation and Interpretation

## [1] "AIC =  9231"

## [1] "BIC =  9267"

The variance components for FEV1 include the standard deviations of the random intercept as “(Intercept)”, slope as “Age” and residual as “Residual” reported below.

Variance Components
	StdDev
(Intercept)	12.429
Age	0.496
Residual	8.696

Longitudinal Process
	Value	Std.Err	z-value
(Intercept)	106.202	0.605	175.524
Age	-1.688	0.028	-59.618
Age_Diag	0.697	0.081	8.635

Event Process
	Value	p-value	Hazard Ratio	Lower	Upper
(Intercept)	-81.815	0.017	0.000	0.000	0.000
Age_Diag	-0.273	0.012	0.761	0.615	0.943
Assoct	0.153	0.142	1.165	0.950	1.428
Assoct.s	-11.658	0.023	0.000	0.000	0.198
log(shape)	2.698	0.000	14.850	6.883	32.038

5.3 Model Inference

5.4 Model Diagnostics

6 Non-linear Joint Model of Longitudinal FEV1(% predicted) and Survival (JM)

6.1 Assumptions

Average evolution of FEV1: A non-linear evolution by cubic natural spline is assumed over time.